Homomorphic white box system and method for using same

ABSTRACT

A method for whitebox cryptography is provided for computing an algorithm  (m,S) with input m and secret S, using one or more white-box encoded operations. The method includes accepting an encoded input c, where c=Enc(P,m); accepting an encoded secret S′, where S′=Enc(P,S); performing one or more operations on the encoded input c and the encoded secret S′ modulo N to obtain an encoded output c′; and decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′= (m,S).

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit of U.S. Provisional Patent Application No. 62/443,926 entitled “CANDIDATE FULLY HOMOMORPHIC WHITE BOX SYSTEM,” by Lex Aaron Anderson, Alexander Medvinsky, and Rafie Shamsaasef, filed Jan. 9, 2017, which application is hereby incorporated by reference herein.

BACKGROUND 1. Field of the Invention

The present invention relates to systems and methods for performing cryptographic operations, and in particular to a system and method for securely performing homomorphic cryptographic operations.

2. Description of the Related Art

The goal of much of cryptography is to allow dissemination of information in such a way that prevents disclosure to unauthorized entities. This goal has been met using cryptographic systems (such as the Advanced Encryption Standard (AES), Triple Data Encryption Standard (TDES), Rivest-Shamir-Adleman (RSA), Elliptic-Curve Cryptography (ECC)) and protocols.

In the systems implementing such cryptographic systems, it is assumed that the attacker only has access to the input and output of the algorithm performing the cryptographic operation, with the actual processing being performed invisibly in a “black box.” For such a model to comply, the black box must provide a secure processing environment. Active research in this domain includes improved and special purpose cryptographic systems (e.g., lightweight block ciphers, authentication schemes, homomorphic public key algorithms), and the cryptanalysis thereof.

While such systems are effective, they are still vulnerable to attack. For example, protocols may be deployed in the wrong context, badly implemented algorithms, or inappropriate parameters may introduce an entry point for attackers.

New cryptanalysis techniques that incorporate additional side-channel information that can be observed during the execution of a crypto algorithm; information such as execution timing, electromagnetic radiation and power consumption. Mitigating such side channel attacks is a challenge, since it is hard to de-correlate this side-channel information from operations on secret keys. Moreover, the platform often imposes size and performance requirements that make it hard to deploy protection techniques.

Further exacerbating the foregoing problems, more applications are being performed on open devices with general purpose processors (e.g. personal computers, laptops, tablets, and smartphones) instead of devices having secure processors.

In response to the foregoing problems, many systems use “white-box” techniques, in which it is assumed that the attacker has full access to the software implementation of a cryptographic algorithm: the binary is completely visible and alterable by the attacker, and the attacker has full control over the execution platform (CPU calls, memory registers, etc.). In such systems, the implementation itself is the sole line of defense.

White-box cryptography was first published by Chow et al. (Stanley Chow, Philip A. Eisen, Harold Johnson, and Paul C. van Oorschot. A white-box DES implementation for DRM applications. In Proceedings of the ACM Workshop on Security and Privacy in Digital Rights Management (DRM 2002), volume 2696 of Lecture Notes in Computer Science, pages 1-15. Springer, 2002, hereby incorporated by reference herein). This addressed the case of fixed key white-box DES implementations. The challenge is to hard-code the DES symmetric key in the implementation of the block cipher. The main idea is to embed both the fixed key (in the form of data but also in the form of code) and random data (instantiated at compilation time) in a composition from which it is hard to derive the original key.

The goal of a white-box attacker is to recover the secret from a white-box implementation. Typically, white-box cryptography is implemented via lookup tables encoded with bijections. Since these bijections are randomly chosen, it is infeasible for an attacker to brute-force the encodings for a randomly chosen secret from a sufficiently large keyspace.

Further, code footprints present a significant problem for typical white-box implementations, which use lookup tables to replace mathematical operations with encoded mathematical operations. For example, if a single operation is to be performed using two one byte (8 bit) numbers, the lookup table will comprise 2⁸ or 256 rows and 256 columns (0 to 255), and will therefore comprise 64K bytes of information that must be stored. Further, computations performed on larger numbers substantially increase storage requirements. For example, if a single operation is to be performed using two 16 bit numbers, the lookup table will comprise 2¹⁶*2¹⁶ rows and columns of 16 bit numbers, which requires more than 8.6 gigabytes of storage. Given that typically more than one cryptographic operation is required and that computations may need to be performed in 32 or 64 bits, it can be seen that classical lookup-table-based white-box implementations are not suited to applications that are based on large integers. Further, while the size of the lookup tables may be reduced by breaking cryptographic computations down into smaller integers, a greater number of lookup tables will be required. For example, it has been estimated that to perform RSA computations in a white-box implementation, several thousand one-byte lookup tables would be required.

What is needed is a way to efficiently perform large integer cryptographic operations offering the advantages of white-box implementations that do not expose secrets to compromise, while minimizing the storage and processing requirements of such implementations.

SUMMARY

To address the requirements described above, the present invention discloses a method and apparatus for computing an algorithm

(m,S) with input m and secret S, using one or more white-box encoded operations. In one embodiment, the method comprises defining a white-box fully-homomorphic key generation function (P,p)←Gen(1^(w)) with public-key P and private-key p that selects random prime numbers p, q, sϵW of similar size, wherein: B={0,1}^(b) is the domain of order b, of the algorithm

; W=[0,1]^(w) is a white-box domain of order w, for w>>b; p>2^(b) is a white-box fully-homomorphic private key, N=pq, k=s(p−1); and P=(N,k) is a white-box fully-homomorphic public key. The method also comprises defining a white-box fully-homomorphic encoding function Enc(P,m):=m^(rk+1)(mod N) that generates a random integer rϵW, then performs an encoding of the input mϵB; and defining a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) that decodes c by computing c modulo p. Finally, the method also comprises accepting an encoded input c, where c=Enc(P,m); accepting an encoded secret S′, where S′=Enc(P,S); performing one or more operations on the encoded input c and the encoded secret S′ modulo N to obtain an encoded output c′; and decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′=

(m,S).

Other embodiments are evidenced by an apparatus having a processor communicatively coupled to a memory storing processor instructions for performing the foregoing operations.

The foregoing allows white-box implementations that are tunable to maximize performance if needed or to achieve security strength as required. It is applicable for direct application to general-purpose program code, thus reducing the expertise required to build and integrate white-box implementations, while also diminishing the incidence of implementation weaknesses through automation.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers represent corresponding parts throughout:

FIGS. 1A and 1B are diagrams of a cryptographic system processing an input message to produce an output message, and its corresponding white-box implementation;

FIG. 2 is a diagram illustrating exemplary operations that can be performed to implement one embodiment of a fully homomorphic white-box implementation (FHWI);

FIG. 3 is a diagram illustrating one embodiment of a key generator for generating a private key p and a public key P;

FIG. 4 is a diagram illustrating one embodiment of a fully homomorphic white-box implementation;

FIG. 5 is a diagram presenting a tabular comparison of processing times for the baseline white-box implementation illustrated in FIG. 1B, and the FHWI implementation shown in FIG. 4;

FIG. 6 is a diagram presenting a tabular comparison of the memory footprint required to implement the baseline white-box implementation illustrated in FIG. 1B and the FHWI implementation illustrated in FIG. 4; and

FIG. 7 is a diagram illustrating an exemplary computer system that could be used to implement elements of the present invention.

DETAILED DESCRIPTION

In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention.

Overview

A fully homomorphic white-box implementation of one or more cryptographic operations is presented below. This method allows construction of white-box implementations from general-purpose code without necessitating specialized knowledge in cryptography, and with minimal impact to the processing and memory requirements for non-white-box implementations. This method and the techniques that use it are ideally suited for securing “math heavy” implementations, such as codecs, that currently do not benefit from white-box security because of memory or processing concerns. Further, the fully homomorphic white-box construction can produce a white-box implementation from general purpose program code, such as C or C++.

In the following discussion, the terms “encoding,” “decoding,” “encoder,” and “decoder,” are used to generally describe such performed operations as being possible to implement in smaller domains. The principles discussed herein may also be applied without loss of generality to larger domains, and in such applications, the operations may be classified as “encrypting” and “decrypting.”

White-Box Cryptographic Systems

A white-box system operates by encoding data elements (such as secret keys) so that they cannot be recovered by an attacker in their cleartext form. A white-box implementation is generated with mathematically altered functions that operate directly on the encoded data elements without decoding them. This guarantees that the secrets remain encoded at all times, thus protecting the implementation against attackers with full access to and control of the execution environment. This is described, for example, in the Chow reference cited above.

FIGS. 1A and 1B are diagrams of a cryptographic system processing an input message to produce an output message, and its corresponding white-box implementation.

As illustrated in FIG. 1A, the algorithm performs functions ƒ₁, ƒ₂ and ƒ_(n) (102A, 102B, and 102N, respectively) when provided with an input and secret S.

In FIG. 1B, each operation ƒ₁, ƒ₂, . . . , ƒ_(n) in an original algorithm

(m,S) with input message m and secret S is encoded as a lookup-table T₁, T₂, . . . , T_(n) (104A, 104B, and 104N, respectively) in the classical white-box implementation of that algorithm. The encodings are generated as two sequences of random bijections, δ₁, δ₂, . . . , δ_(n+1) that are applied to the inputs and output of each operation, where ρ(S) represents an encoded secret (e.g. a secret key), which is either linked statically or provided dynamically to the white-box implementation.

In the white-box implementation shown in FIG. 1B this is implemented by applying bijections δ₁ and ρ(S) as an input to lookup table T₁ to obtain an intermediate output, applying the intermediate output and ρ(S) to lookup table T₂ to produce a second intermediate output, then providing the second intermediate output and ρ(S) to lookup table T₃ to produce output δ_(n+1) ⁻¹(⋅). Lookup table T₁ inverts the bijection δ₁ of the input by δ₁ ⁻, inverts the bijection ρ of S(ρ(S)) by ρ₁ ⁻¹, applies ƒ₁ and then applies bijection δ₂ to produce the first intermediate output. Similarly, lookup table T₂ inverts the bijection δ₂ of the first intermediate input by δ₂ ⁻¹, inverts the bijection ρ of S(ρ(S)) by ρ₂ ⁻¹, applies ƒ₂ and then applies bijection δ₃ to produce the first intermediate output. Generally, final lookup table T_(n) inverts the bijection δ_(n) of the n−1^(th) intermediate input by δ_(n) ⁻¹, inverts the bijection ρ of S(ρ(S)) by ρ_(n) ⁻¹, applies ƒ_(n) and then applies bijection δ_(n+1) to produce the intermediate output δ_(n+1) ⁻¹(⋅).

White-box implementations are usually of cryptographic primitives (such as the advanced encryption standard or AES) and chains of cryptographic primitives used in cryptographic protocols (such as elliptic curve Diffie-Hellman key exchange, or ECDHE). Designing and coding these implementations requires specialist knowledge of white-box cryptography to attain the best possible security properties. Just as importantly, integration with white-box implementations also requires a degree of specialist knowledge to avoid common pitfalls that can negate the white-box security.

Public Key Cryptography

Public key encryption schemes use a pair of keys: a public key which may be disseminated widely, and a private key which are known only to the owner. The message is encrypted according to the public key, and the public key is publicly shared. However, decrypting the message requires the private key, which is only provided to authorized recipients of the encrypted data. This accomplishes two functions: authentication and encryption. Authentication is accomplished because the public key can be used to verify that a holder of the paired private key sent the message. Encryption is accomplished, since only a holder of the paired private key can decrypt the message encrypted with the public key.

A public-key encryption scheme is a triple (Gen, Enc, Dec), with a probabilistic-polynomial-time (PPT) key-pair generator algorithm Gen, PPT encryption algorithm Enc and PPT decryption algorithm Dec, such that for any public/private key-pair (e,d)←Gen(

) and all messages m of length

it holds that m=Dec(d,Enc(e,m)).

Homomorphic Cryptographic Operations

Fully homomorphic encryption schemes preserve underlying algebraic structure, which allows for performing operations in an encrypted domain without the need for decryption, as described in “On data banks and privacy homomorphisms,” by Ronald L Rivest, L Adleman, and M L Dertouzos, Foundations of Secure Computation, 32(4):169-178, 1978, which is hereby incorporated by reference herein.

As used herein, a fully homomorphic encryption scheme is an encryption scheme with the following property: Given two encryption operations Enc(e,m₁) and Enc(e,m₂), where m₁ and m₂ are two messages encrypted with a chosen public key e, one can efficiently and securely compute c=Enc(e,m₁⊙m₂)=Enc(e,m₁)⊙Enc(e,m₂) without revealing m and m₂, such that Dec(d,c)=m₁⊙m₂, wherein the operation ⊙ is multiplication or addition.

Thus, homomorphic cryptography is a form of cryptography that permits computation on encrypted data or ciphertexts, which, when decrypted, provides the same result as would have been provided if the computations were performed on the unencrypted or plaintext. Hence, homomorphic cryptography permits the performance of one or more cryptographic operations, while not exposing the secrets used in such operations.

White-Box Fully Homomorphic Cryptographic Processing

A number-theoretic white-box encoding scheme that is suited to arithmetic operations in common use in software applications is presented below. The encoding scheme is based on Fermat's Little Theorem, which states that if p is a prime number and if α is any integer not divisible by p, then α^(p−1)−1 is divisible by p.

A white-box fully homomorphic encoding scheme (WBFHE) can be defined as follows. Let B=0,1^(b), b≥8 be the integral domain of the arithmetic operations in the original algorithm (e.g. the one or more operations depicted in FIG. 1A). The term b represents the order of the integral domain. For example, if b=8, integral domain B consists of 2⁸ possible values. Further, let W=0,1^(w), w>>b be the white-box domain, such that Enc: W×B→W is a WBFHE encoding and Dec: W×W→B is a WBFHE decoding. The term w refers to the order of the white-box domain. For example, if w=1000, the white-box domain includes 2¹⁰⁰⁰ possible values.

Three functions (Gen, Enc, and Dec) are defined. The Gen function selects three random prime integers p, q, sϵW of similar size (e.g. same order of magnitude), where p>2^(b) is the private key. Let N=pq and let k=s(p−1) such that P=(N,k) is a public key, where keypair generation is denoted by:

(P,p)←Gen(1^(w))  Equation (1)

The Enc function generates a random integer rϵW, then an encoding of input message mϵB is defined as:

c=Enc(P,m):=m ^(rk+1)(mod N)  Equation (2)

The decoding function decodes c to recover an encoded message m by computing c modulo p as follows:

m=Dec(p,c)=c(mod p)  Equation (3)

The order w of the white-box domain W is a parameter that can be adjusted or tuned to increase or decrease security to obtain the desired level of performance from the white box implementation. If w is sufficiently large, then WBFHE can be considered an encryption scheme with semantic security, as described below.

The foregoing WBFHE is multiplicatively and additively homomorphic. These properties can be validated as follows:

Let (P,p)←Gen(1^(w)) and choose m₁, m₂ϵB. If the following encryptions are computed:

c ₁ =Enc(P,m ₁)=m ^(r) ¹ ^(k+1)(mod N)=m ₁ ^(r) ¹ ^(s(p−1)+1)(mod N)  Equation (4)

c ₂ =Enc(P,m ₂)=m ^(r) ² ^(k+1)(mod N)=m ₂ ^(r) ² ^(s(p−1)+1)(mod N)  Equation (5)

It can be shown that:

m ₁ m ₂ =Dec(p,c ₁ c ₂)=c ₁ c ₂(mod p)  Equation (6)

and

m ₁ +m ₂ =Dec(p,c ₁ +c ₂)=c ₁ +c ₂(mod p)  Equation (7)

For example, consider a small integer domain for purposes of illustration where:

message one=m₁=8

message two=m₂=11

private key=p=101

first random integer=r₁=219

second random integer=r₂=112 and

third random prime number s=97

Substituting these values into Equations (6) and (7), respectively yields Equations (8) and (9) below:

4250=Enc(P,m ₁)=m ^(r) ¹ ^(k+1)(mod N)=m ₁ ^(219*97(101−1)+1)(mod 8989)  Equation (8)

2132=Enc(P,m ₂)=m ^(r) ² ^(k+1)(mod N)=m ₂ ^(112*97(101−1)+1)(mod 8989)  Equation (9)

Homomorphic addition can be shown because:

m ₁ +m ₂ =Dec(p,m ₁ +m ₂)=4250+2132(Mod 101)  Equation (10)

8+11=6382(mod 101)

19=19  Equation (11)

Homomorphic multiplication can be shown because:

m ₁ *m ₂ =Dec(p,m ₁ *m ₂)=4250*2132(Mod 101)  Equation (12)

8*11=9061000(mod 101)

88=88  Equation (13)

Further, since the foregoing white-box implementation is both multiplicatively and additively homomorphic, it is fully homomorphic.

FIG. 2 is a diagram illustrating exemplary operations that can be performed to implement one embodiment of a fully homomorphic white-box implementation. FIG. 2 will be discussed in conjunction with FIGS. 3 and 4, which depict one embodiment of a key pair generator 300 and a fully homomorphic white-box implementation 400 corresponding to the processing system depicted in FIG. 1A, respectively.

Turning first to FIG. 2, block 210 encodes an input message m to compute an encoded input c. This can be accomplished using encoder 404 depicted in FIG. 4 according to Equation (2) above.

The private key p and public key P=(N,k) is generated by the key pair generator 300 depicted in FIG. 3, which comprises a random number generator (RNG) 302 for generating random prime numbers p, q and s such that p, q, sϵ0,1^(w). The key generator 300 provides the random prime number p as the private key and a public key P=(N,k) computed as a tuple of N by element 308, and N is a product of random prime numbers p and q (as computed by multiplication element 304) and k=s(p−1), as computed by element 306. The factor r in Equation (2) is a random integer that need not be prime.

Turning again to FIG. 2, block 212 encodes a secret S to compute an encoded secret S′. This can be accomplished by using encoder 404′ depicted in FIG. 4 according to Equation (2), with the Enc function is performed on secret S. The factor r used to compute encoded secret S′ is a random integer that need not be prime. This second random integer may be generated by a random integer generator in the encoder 404′ illustrated in FIG. 4, or may be generated by the random number generator 302 of the key pair generator 300 illustrated in FIG. 3 and provided to the encoder 404′.

Returning to FIG. 2, the encoded input c and the encoded secret S′ are transmitted from the transmitter 202 to the receiver 204, as shown in block 214. The receiver 204 accepts the transmitted encoded input c and the transmitted encoded secret S′, as shown in block 216, and performs one or more cryptographic operations according to the encoded input message c and the encoded secret S′ to compute an encoded output c′. These operations are performed modulo N. The resulting encoded output c′ is then decoded using private key p to recover the output message m′ as shown in block 222. The decryption may be performed, for example by generating the output message m′ according to the Dec function of Equation (3) applied to c′, or:

m′=Dec(p,c′)=c′(mod p)  Equation (14)

Note that since all of the operations ƒ′₁, ƒ′₂ and ƒ′_(n) are performed on encoded data (e.g. the data is not decoded until all of the operations ƒ′₁, ƒ′₂ and ƒ′_(n) have been performed), where it is difficult for an attacker to use the intermediate results of such calculations to divine the value of the secret S. This property is made possible by the homomorphic character of the white-box processing system, which permits the operations to be performed on the encoded data rather than requiring the data be decoded before the operations are performed.

Further, as described above, the order w of the white-box domain W can be selected to provide the desired level of security. In other words, the larger the domain from which the random prime integers p, q, s and random integers rare chosen from, the more security is provided. For example, if the numbers p, q and s are of at least w bits in size where w is much greater than (>>) b where b is the order of the integral domain B of the original operations ƒ₁, ƒ₂ and ƒ_(n) (i.e., if the input message m may comprise an integer of at least b bits in size), semantic cryptographic security may be obtained.

The operations depicted in FIG. 1A comprise the serial performance of functions ƒ₁, ƒ₂, and ƒ_(n) (102A, 102B, and 102N, respectively) using secret S. The same functions are performed in the fully homomorphic white-box implementation 400 depicted in FIG. 4, but are performed in the white-box domain W, a difference of functionality that is indicated by ƒ′₁, ƒ′₂ and ƒ′_(n) (102A′, 102B′, and 102N′, respectively).

For exemplary purposes, consider a case where the one or more operations comprise ƒ₁ and ƒ₂, with ƒ₁ and ƒ₂ defined as follows for inputs (x,y) as follows:

ƒ₁(x,y)=x+y  Equation (15)

ƒ₂(x,y)=xy  Equation (16)

wherein operation ƒ′₁ computes the sum of the encoded input message m and encoded secret S′ modulo N to compute intermediate output, operation ƒ′₂ computes the product of the intermediate output and the encoded secret S′ modulo N to compute the output, which is the encoded output c′. Hence, each operation ƒ₁, ƒ₂: BλB→B is implemented in the white-box domain ƒ′₁, ƒ′₂:WλW→W. Further, as demonstrated below, since the cryptographic operations are homomorphic in both addition and multiplication, they are fully homomorphic.

While the foregoing example uses two functions ƒ₁ and ƒ₂, a greater number of functions may be used while maintaining the homomorphic character of the implementation. Since many other functional relationships can be described as various combinations of addition or multiplication, a wide variety of functions can be implemented with such combinations.

Note that since all of the operations ƒ′₁, ƒ′₂ and ƒ′_(n) are performed on encoded data (e.g. the data is not decrypted until all of the operations ƒ′₁, ƒ′₂ and ƒ′_(n) have been performed), it is difficult for an attacker to use the intermediate results of such calculations to divine the value of the secret S. This property is made possible by the homomorphic character of the white-box processing system, which permits the operations to be performed on the encoded data rather than requiring the data be decrypted before the operations are performed.

Functional Allocation Among Elements

Importantly, the process of key generation, the encoding of the secret and decoding of any data is performed external to the white-box implementation 400 and in a secure environment. Further, if the private key (p) is to be disseminated to the receiver of the message for use, such dissemination must be performed securely.

For example, if key generation were performed in the white-box implementation 400 itself, an attacker could intercept the generated private key component (p) and use it to decode any encoded data in the white-box implementation 400, including the encoded secret (S). For example, if the white-box implementation is of RSA and the secret S represents the RSA private key, then an attacker could simply use S in non-white-box RSA to decrypt the input, bypassing the white-box implementation 400 entirely. Further, with respect to encoding the secret S (performed by encoder 404′), such encoding of the secret S requires knowledge of the unencoded secret S. Therefore, if the encoding were performed on an insecure device, the secret S would be exposed because it is an input to the encoding operation. Since the unencoded secret could be used in a non-white-box version of the cryptosystem to carry out the original cryptographic operation, the protection afforded by the white-box implementation 400 would be negated. Finally, with respect to the decoding of data, such decoding requires knowledge of the private key (p), and an attacker with knowledge of the private key can decode the encoded secret S and use this in a non-white-box version of the cryptosystem to carry out the original cryptographic operation.

Semantic Security

Encryptions can be described as having a property of indistinguishability. This is described in “A uniform-complexity treatment of encryption and zero knowledge,” by Oded Goldreich, Journal of Cryptology: The Journal of the International Association for Cryptologie Research (IACR), 6(1):21-53, 1993, which is hereby incorporated by reference. The indistinguishability property states that it is infeasible to find pairs of messages for which an efficient test can distinguish corresponding encryptions.

An algorithm

may be said to distinguish the random variables R_(n) and S_(n) if

behaves substantially differently when its input is distributed as R_(n) rather than as Sn. Without loss of generality, it suffices to ask whether Pr[

(R_(n))=1] and Pr[

(S_(n))=1] are substantially different.

An encryption scheme (Gen, Enc, Dec) has indistinguishable encryptions if for every polynomial-time random variable {T_(n)=X_(n)Y_(n)Z_(n)}_(nϵN) with |X_(n)|=|Y_(n)|, every probabilistic polynomial-time algorithm

, every constant c>0 and all sufficiently large n, and a fixed P←Gen{1^(n)},

$\begin{matrix} {{{{\Pr \left\lbrack {{\left( {Z_{n},{{Enc}\left( {p,X_{n}} \right)}} \right)} = 1} \right\rbrack} - {\Pr\left\lbrack {{\left( {Z_{n},{{Enc}\left( {P,Y_{n}} \right)}} \right)} = 1} \right.}} < \frac{1}{n_{c}}}} & {{Equation}\mspace{14mu} (16)} \end{matrix}$

The probability in the above terms is taken over the probability space underlying T_(n) and the internal coin tosses of the algorithms Gen, Enc and

.

It has also been shown that semantic security is equivalent to indistinguishability of encryptions, which allows proof that a WBFHE described above are semantically secure for a sufficiently large white box domain W. See, for example “Probabilistic encryption & how to play mental poker keeping secret all partial information,” by Shafi Goldwasser and Silvio Micali, STOC '82 Proceedings of the Fourteenth Annual ACM symposium on Theory of computing, pages 365-377, 1982, which is hereby incorporated by reference herein. The proof is provided as follows:

If we choose a random mϵB and public key P←Gen{1^(n)}, and suppose that an encryption process Enc for two encryptions c₁=Enc(P,m) and c₂=Enc(P,m) is not probabilistic. Then c₁=c₂. But since the encryption process Enc chooses rϵW at random for each encryption,

${{\Pr \left\lbrack {c_{1} = c_{2}} \right\rbrack} = \frac{1}{2^{w}}},$

which is negligible. This is a contradiction. Hence, Enc is probabilistic, and thus if w were sufficiently large, an adversary without knowledge of r has a negligible advantage of using knowledge of Enc to compute the same ciphertext as an oracle implementation of Enc(P,m).

A connection can also be shown between the WBFHE and the integer factorization problem, where it is noted that no efficient (polynomial time) integer factorization algorithm (as discussed in “Number Theory for Computing,” by Song Y Yan, Springer Berlin Heidelberg, 2002, hereby incorporated by reference herein. If a PPT algorithm

can factor N=pq or k=s(p−1), then there exists a PPT algorithm

that can invert Enc(P,m). This is apparent because the WBFHE private key p is a prime factor of N and also (p−1) is a factor of k.

Exemplary Applications

The foregoing principles can be applied to any cryptographic function having one or more cryptographic operations, including digital signatures and their use, encryption and decryption. For exemplary purposes, an application to the decryption of an RSA encrypted message is described. In this case, the algorithm

is an RSA decryption algorithm. Further, the accepted encoded message is c=Enc(P,M), wherein M is an RSA encrypted version of the input message m encoded with the public key P, and the accepted encoded secret is S′=Enc(P,RSAPVK), wherein RSAPVK is the RSA private key encoded with the public key P. In this case, the one or more cryptographic operations comprise RSA decrypt operations on the encoded input c and the encoded secret S′ to compute the encoded output c′. Hence, the RSA decrypt operations operate on the encrypted version of the input message M and the encoded version of the RSA private key, RSAPVK to produce an encoded output c′ without exposing the RSA private key, and the original message m can be recovered using the private key p according to m=Dec(p,c′).

Another exemplary application of the foregoing principles involves the decryption of programs that have been compressed, for example according to an MPEG (motion pictures working guild) standard. Typically, a media program is compressed, the compressed version of the media program encrypted and thereafter transmitted. The compressed and encrypted media program is received by the receiver, decrypted, then decompressed. Once decrypted, the media program is exposed in compressed form and is vulnerable to compromise before the decompression process. Further, the media program is exposed in compressed form which is of smaller size and can be more easily disseminated to unauthorized viewers.

Using the foregoing principles, the media program is also compressed according to the MPEG standard, and thereafter encoded or encrypted before dissemination. The media program may then be decompressed using a homomorphic implementation using one or more operations. However, the resulting decompressed media program is still in encoded or encrypted form, and is unviewable until the decoding step is applied. At this point, even if the media program were compromised, it would be of much larger size and more difficult to disseminate to unauthorized viewers.

Test Results

Tests were performed with a prototype fully homomorphic white-box implementation (FHWI) written in C++ consisting of 10,000 iterated additions and multiplications. The baseline was computed using built-in 64 bit integral types. An ESCP VLI library was used for the FHWI large integer operations.

FIG. 5 is a diagram presenting a tabular comparison of processing times for the baseline white-box implementation illustrated in FIG. 1B, and the FHWI implementation shown in FIG. 6. Note that the FHWI can take from 3.2 to 144 more time to perform an iteration than the baseline white-box implementation. Note also that the performance penalty is a function of w, so w may be chosen to obtain a desired security level, while minimizing processing penalties.

FIG. 6 is a diagram presenting a tabular comparison of the memory footprint required to implement the baseline white-box implementation illustrated in FIG. 1B and the FHWI implementation illustrated in FIG. 6. Note that the FHWI results in significant footprint reductions (the memory footprint required for the implementation is reduced by a factor of about 64 (w=1024) to 114 (w=128).

Hardware Environment

FIG. 7 is a diagram illustrating an exemplary computer system 700 that could be used to implement elements of the present invention, including the transmitter 202, receiver 204, processor 206, encoder 404, 404′ and decryptor 406. The computer 702 comprises a general-purpose hardware processor 704A and/or a special purpose hardware processor 704B (hereinafter alternatively collectively referred to as processor 704) and a memory 706, such as random-access memory (RAM). The computer 702 may be coupled to other devices, including input/output (I/O) devices such as a keyboard 714, a mouse device 716 and a printer 728.

In one embodiment, the computer 702 operates by the general-purpose processor 704A performing instructions defined by the computer program 710 under control of an operating system 708. The computer program 710 and/or the operating system 708 may be stored in the memory 706 and may interface with the user and/or other devices to accept input and commands and, based on such input and commands and the instructions defined by the computer program 710 and operating system 708 to provide output and results.

Output/results may be presented on the display 722 or provided to another device for presentation or further processing or action. In one embodiment, the display 722 comprises a liquid crystal display (LCD) having a plurality of separately addressable pixels formed by liquid crystals. Each pixel of the display 722 changes to an opaque or translucent state to form a part of the image on the display in response to the data or information generated by the processor 704 from the application of the instructions of the computer program 710 and/or operating system 708 to the input and commands. Other display 722 types also include picture elements that change state in order to create the image presented on the display 722. The image may be provided through a graphical user interface (GUI) module 718A. Although the GUI module 718A is depicted as a separate module, the instructions performing the GUI functions can be resident or distributed in the operating system 708, the computer program 710, or implemented with special purpose memory and processors.

Some or all of the operations performed by the computer 702 according to the computer program 710 instructions may be implemented in a special purpose processor 704B. In this embodiment, some or all of the computer program 710 instructions may be implemented via firmware instructions stored in a read only memory (ROM), a programmable read only memory (PROM) or flash memory within the special purpose processor 704B or in memory 706. The special purpose processor 704B may also be hardwired through circuit design to perform some or all of the operations to implement the present invention. Further, the special purpose processor 704B may be a hybrid processor, which includes dedicated circuitry for performing a subset of functions, and other circuits for performing more general functions such as responding to computer program instructions. In one embodiment, the special purpose processor is an application specific integrated circuit (ASIC).

The computer 702 may also implement a compiler 712 which allows an application program 710 written in a programming language such as COBOL, C++, FORTRAN, or other language to be translated into processor 704 readable code. After completion, the application or computer program 710 accesses and manipulates data accepted from I/O devices and stored in the memory 706 of the computer 702 using the relationships and logic that was generated using the compiler 712.

The computer 702 also optionally comprises an external communication device such as a modem, satellite link, Ethernet card, or other device for accepting input from and providing output to other computers.

In one embodiment, instructions implementing the operating system 708, the computer program 710, and/or the compiler 712 are tangibly embodied in a computer-readable medium, e.g., data storage device 720, which could include one or more fixed or removable data storage devices, such as a zip drive, floppy disc drive 724, hard drive, CD-ROM drive, tape drive, or a flash drive. Further, the operating system 708 and the computer program 710 are comprised of computer program instructions which, when accessed, read and executed by the computer 702, causes the computer 702 to perform the steps necessary to implement and/or use the present invention or to load the program of instructions into a memory, thus creating a special purpose data structure causing the computer to operate as a specially programmed computer executing the method steps described herein. Computer program 710 and/or operating instructions may also be tangibly embodied in memory 706 and/or data communications devices 730, thereby making a computer program product or article of manufacture according to the invention. As such, the terms “article of manufacture,” “program storage device” and “computer program product” or “computer readable storage device” as used herein are intended to encompass a computer program accessible from any computer readable device or media.

Of course, those skilled in the art will recognize that any combination of the above components, or any number of different components, peripherals, and other devices, may be used with the computer 702.

Although the term “computer” is referred to herein, it is understood that the computer may include portable devices such as cellphones, portable MP3 players, video game consoles, notebook computers, pocket computers, or any other device with suitable processing, communication, and input/output capability.

CONCLUSION

This concludes the description of the preferred embodiments of the present invention. The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching.

It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. The above specification, examples and data provide a complete description of the manufacture and use of the apparatus and method of the invention. Since many embodiments of the invention can be made without departing from the scope of the invention, the invention resides in the claims hereinafter appended. 

What is claimed is:
 1. A method of computing an algorithm

(m,S) with input m and secret S, using one or more white-box encoded operations, comprising: defining a white-box fully-homomorphic key generation function (P,p)←Gen(1^(w)) with public-key P and private-key p that selects random prime numbers p, q, sϵW of similar size, wherein: B={0,1}^(b) is the domain of order b, of the algorithm

; W={0,1}^(w) is a white-box domain of order w, for w>>b; p>2^(b) is a white-box fully-homomorphic private key; N=pq; k=s(p−1); P=(N,k) is a white-box fully-homomorphic public key; defining a white-box fully-homomorphic encoding function Enc(P,m):=m^(rk+1)(mod N) that generates a random integer rϵW, then performs an encoding of the input mϵB; defining a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) that decodes c by computing c modulo p; accepting an encoded input c, where c=Enc(P,m); accepting an encoded secret S′, where S′=Enc(P,S); performing one or more operations on the encoded input c and the encoded secret S′ modulo N to obtain an encoded output c′; and decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′=

(m,S).
 2. The method of claim 1, wherein the algorithm

is comprises a decryption algorithm including at one of an Rivest-Shamir-Aldeman (RSA) algorithm, an elliptic curve cryptography (ECC) algorithm, an advanced encryption standard (AES) algorithm, and a triple data standard (TDES) algorithm.
 3. The method of claim 2, wherein: the algorithm

is an RSA decryption algorithm RSADecrypt; the accepted encoded message is c=Enc(P,M), wherein M=RSAEncrypt(RSAPLK,m) is an RSA encrypted version of the input message m encoded with the white-box fully-homomorphic public key P, where (RSAPVK,RSAPLK) is a RSA private/public keypair corresponding to the RSADecrypt and RSAEncrypt algorithms; the accepted encoded secret is S′=Enc(P,RSAPVK), wherein RSAPVK is the RSA private key encoded with the white-box fully-homomorphic public key P; the one or more operations comprise RSADecrypt implementation, with encoded input c and the encoded secret S′ to compute the encoded output c′; and decoding the encoded output c′ with the private key p to recovers the output message m′ according to m′=Dec(p,c′).
 4. The method of claim 1, wherein w is selected for semantic security.
 5. The method of claim 1, further comprising: securely encoding the input message m according to c=Enc(P,m); and securely encoding the secret S according to S′=Enc(P,S).
 6. An apparatus for computing an algorithm

(m,S) with input m and secret S, using one or more white-box encoded operations, comprising: means for defining a white-box fully-homomorphic key generation function (P,p)←Gen(1^(w)) with public-key P and private-key p that selects random prime numbers p, q, sϵW of similar size, wherein: B={0,1}^(b) is the domain of order b, of the algorithm

; W={0,1}^(w) is a white-box domain of order w, for w>>b; p>2^(b) is a white-box fully-homomorphic private key; N=pq; k=s(p−1); P=(N,k) is a white-box fully-homomorphic public key; means for defining a white-box fully-homomorphic encoding function Enc(P,m):=m^(rk+1)(mod N) that generates a random integer rϵW, then performs an encoding of the input mϵB; means for defining a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) that decodes c by computing c modulo p; a processor, a memory, communicatively coupled to the processor, the memory storing processor instructions comprising processor instructions for: accepting an encoded input c, where c=Enc(P,m); accepting an encoded secret S′, where S′=Enc(P,S); performing one or more operations on the encoded input c and the encoded secret S′ modulo N to obtain an encoded output c′; and decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′=

(m,S).
 7. The apparatus of claim 6, wherein the algorithm

is comprises a decryption algorithm including at one of an Rivest-Shamir-Aldeman (RSA) algorithm, an elliptic curve cryptography (ECC) algorithm, an advanced encryption standard (AES) algorithm, and a triple data standard (TDES) algorithm.
 8. The apparatus of claim 7, wherein: the algorithm

is an RSA decryption algorithm RSADecrypt; the accepted encoded message is c=Enc(P,M), wherein M=RSAEncrypt(RSAPLK,m) is an RSA encrypted version of the input message m encoded with the white-box fully-homomorphic public key P, where (RSAPVK,RSAPLK) is a RSA private/public keypair corresponding to the RSADecrypt and RSAEncrypt algorithms; the accepted encoded secret is S′=Enc(P,RSAPVK), wherein RSAPVK is the RSA private key encoded with the white-box fully-homomorphic public key P; the one or more operations comprise RSADecrypt implementation, with encoded input c and the encoded secret S′ to compute the encoded output c′; and decoding the encoded output c′ with the private key p to recovers the output message m′ according to m′=Dec(p,c′).
 9. The apparatus of claim 6, wherein w is selected for semantic security.
 10. The method of claim 6, further comprising: securely encoding the input message m according to c=Enc(P,m); and securely encoding the secret S according to S′=Enc(P,S).
 11. A method of computing an algorithm

(m,S) with input m and secret S, using one or more white-box encoded operations, comprising accepting an encoded input c, where c=Enc(P,m); accepting an encoded secret S′, where S′=Enc(P,S); performing one or more operations on the encoded input c and the encoded secret S′ modulo N to obtain an encoded output c′; and decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′=

(m,S). wherein: a white-box fully-homomorphic key generation function (P,p)←Gen(1^(w)) is defined with public-key P and private-key p that selects random prime numbers p, q, sϵW of similar size, wherein: B={0,1}^(b) is the domain of order b, of the algorithm

; W={0,1}^(w) is a white-box domain of order w, for w>>b; p>2^(b) is a white-box fully-homomorphic private key; N=pq; k=s(p−1); P=(N,k) is a white-box fully-homomorphic public key; a white-box fully-homomorphic encoding function Enc(P,m):=m^(rk+1)(mod N) is defined that generates a random integer rϵW, then performs an encoding of the input mϵB; a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) is defined that decodes c by computing c modulo p.
 12. The method of claim 11, wherein the algorithm A is comprises a decryption algorithm including at one of an Rivest-Shamir-Aldeman (RSA) algorithm, an elliptic curve cryptography (ECC) algorithm, an advanced encryption standard (AES) algorithm, and a triple data standard (TDES) algorithm.
 13. The method of claim 12, wherein: the algorithm

is an RSA decryption algorithm RSADecrypt; the accepted encoded message is c=Enc(P,M), wherein M=RSAEncrypt(RSAPLK,m) is an RSA encrypted version of the input message m encoded with the white-box fully-homomorphic public key P, where (RSAPVK,RSAPLK) is a RSA private/public keypair corresponding to the RSADecrypt and RSAEncrypt algorithms; the accepted encoded secret is S′=Enc(P,RSAPVK), wherein RSAPVK is the RSA private key encoded with the white-box fully-homomorphic public key P; the one or more operations comprise RSADecrypt implementation, with encoded input c and the encoded secret S′ to compute the encoded output c′; and decoding the encoded output c′ with the private key p to recovers the output message m′ according to m′=Dec(p,c′).
 14. The method of claim 11, wherein w is selected for semantic security.
 15. The method of claim 11, further comprising: securely encoding the input message m according to c=Enc(P,m); and securely encoding the secret S according to S′=Enc(P,S). 